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A finite superposition of energy eigenrays: exact phases, laws and return times

Physical scope

A finite, exact model computation for a declared isolated system: the evolution of a nonzero finite superposition of orthogonal energy eigenrays, the resulting trigonometric-polynomial form of the outcome laws of bounded assigned sharp observables, the exact return-time criterion for the ray, the two-energy period, and the stationary case. The eigenvalues are model parameters; the item asserts no apparatus, no detection model and no empirical value, and no finite set of energies is identified with a physical spectrum.

Example

Let $H \neq {0}$ be the state space of a declared isolated system of the framework (The state space of the framework: preparations, pure rays and density operators), let $(H_{o p}, u_{H})$ be its assigned energy observable with spectral projection-valued measure $E_{H}$ and flow $U (t) = \exp (- i t H_{o p} / ℏ)$ (Assigned energy observable, unitary flow and the two pictures of autonomous evolution), and let $u_{1}, \dots, u_{N}$ be a finite orthonormal family of eigenvectors of $H_{o p}$ with real eigenvalues $λ_{1}, \dots, λ_{N}$ : $H_{o p} u_{n} = λ_{n} u_{n}$ for $n = 1, \dots, N$ . Fix coefficients $c_{n} \in C$ and the nonzero vector $ψ_{0} = \sum_{n = 1}^{N} c_{n} u_{n}, S : = {n : c_{n} \neq 0} .$ For a nonzero vector $ϕ$ , write $μ_{[ϕ]}^{A}$ for the outcome law of the pure preparation $[ϕ]$ , computed on the unit representative $ϕ / ∥ ϕ ∥$ as in Assigned sharp observables and the outcome law of the framework clause (b) and State vectors, unit vectors, rays, normalization and phases of a Hilbert space. Then, with $ψ (t) = U (t) ψ_{0}$ :

(i) $ψ (t) = \sum_{n \in S} c_{n} e^{- i λ_{n} t / ℏ} u_{n}$ for every $t$ , and $∥ ψ (t) ∥ = ∥ ψ_{0} ∥$ ; if $S$ has one element, or more generally if all $λ_{n}$ with $n \in S$ coincide, the ray $[ψ (t)]$ is constant.

(ii) For every bounded assigned sharp observable $(A, u_{A})$ and every Borel set $B \subseteq R$ , the outcome law in the preparation $[ψ (t)]$ is $μ_{[ψ (t)]}^{A} (B) = \frac{1}{∥ ψ_{0} ∥^{2}} \sum_{m, n \in S} c_{n} \overline{c_{m}} e^{- i (λ_{n} - λ_{m}) t / ℏ} ⟨ E_{A} (B) u_{n}, u_{m} ⟩ .$ Thus the probability is a finite trigonometric polynomial in $t$ with frequencies $(λ_{n} - λ_{m}) / ℏ$ .

(iii) For $T > 0$ the following are equivalent: (a) $[ψ (t + T)] = [ψ (t)]$ for every $t$ ; (b) $e^{- i λ_{n} T / ℏ}$ is independent of $n \in S$ ; (c) $(λ_{m} - λ_{n}) T / ℏ \in 2 π Z$ for all $m, n \in S$ .

(iv) Consequently a positive return time exists if and only if there are $ω_{0} > 0$ and integers $k_{n}$ ( $n \in S$ ) with $(λ_{n} - λ_{n_{0}}) / ℏ = ω_{0} k_{n}$ for all $n \in S$ , where $n_{0}$ is any fixed element of $S$ ; and then $T = 2 π / ω_{0}$ is a return time.

(v) Two-energy case $N = 2$ , $λ_{1} \neq λ_{2}$ , $c_{1} c_{2} \neq 0$ : the positive return times are exactly $T = 2 π ℏ k / ∣ λ_{1} - λ_{2} ∣$ , $k = 1, 2, 3, \dots$ ; the least return time is $2 π ℏ / ∣ λ_{1} - λ_{2} ∣$ , which is the period $2 π / ∣ ω ∣$ of the precession example with $ω = (λ_{2} - λ_{1}) / ℏ$ .

(vi) In the stationary case of (i), every outcome law of every assigned sharp observable is independent of $t$ : $μ_{[ψ (t)]}^{A} (B) = μ_{[ψ (0)]}^{A} (B)$ .

The computation is a model-internal one: the eigenvalues are parameters of the assigned observable, and no identification with a physical spectrum, apparatus or measured value is asserted.

Facts & Assumptions

Given: A nonzero complex Hilbert space $H$ ; the assigned energy observable $(H_{o p}, u_{H})$ with flow $U (t)$ ; a finite orthonormal family $u_{1}, \dots, u_{N}$ of eigenvectors with real eigenvalues $λ_{n}$ ; coefficients $c_{n}$ and the nonzero vector $ψ_{0}$ ; $S = {n : c_{n} \neq 0}$ ; the evolved vectors $ψ (t) = U (t) ψ_{0}$ ; a bounded assigned sharp observable $(A, u_{A})$ with spectral measure $E_{A}$ ; a Borel set $B$ ; the Axiom of Countable Choice and the Axiom of Choice, the choice hypotheses of the cited spectral suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice).

[F1]

Eigenrays evolve by phases. If $H_{o p} u = λ u$ with $λ \in R$ , then $U (t) u = e^{- i λ t / ℏ} u$ for every $t$ , and the flow is linear with $∥ ψ (t) ∥ = ∥ ψ_{0} ∥$ (Energy conservation, conserved sharp observables and the stationary eigenrays clause (c), The Schroedinger-picture flow: unitarity, domain invariance and the domain-qualified equation clause (a), Conserved spectral probability laws for commuting observables and stationary eigenrays clause (d)).

[F2]

Outcome law for a pure preparation. $μ_{ψ}^{A} (B) = ∥ E_{A} (B) ψ ∥^{2} = ⟨ E_{A} (B) ψ, ψ ⟩$ for a unit vector, and the number depends only on the ray $[ψ]$ ; the inner product is linear in the first argument and conjugate-linear in the second (Assigned sharp observables and the outcome law of the framework clause (b), State vectors, unit vectors, rays, normalization and phases of a Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument).

[F3]

Complex exponential algebra. For real $θ, ϕ$ : $e^{i (θ + ϕ)} = e^{i θ} e^{i ϕ}$ , $∣ e^{i θ} ∣ = 1$ and $\overline{e^{i θ}} = e^{- i θ}$ ; for complex $z, w$ , $\overline{z w} = \overline{z} \overline{w}$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive). Moreover $\ker (\exp) = 2 π i Z$ , so for real $μ$ one has $e^{- i μ} = 1$ exactly for $μ \in 2 π Z$ ( $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ).

[F4]

Orthonormality. $⟨ u_{n}, u_{m} ⟩ = δ_{n m}$ , and every $ψ_{0}$ as above is nonzero exactly when $S$ is nonempty (Real and complex inner product spaces, with the inner product linear in the first argument, The induced length is a norm).

Verification

technique · direct; expand the evolved vector along the eigenfamily, evaluate the pairing term by term and read the return-time condition off the phases

1.1givenF1F3algebra

[Evolution of the superposition] By [F1] applied to each eigenvector, $U (t) u_{n} = e^{- i λ_{n} t / ℏ} u_{n}$ ; linearity of $U (t)$ gives $ψ (t) = \sum_{n = 1}^{N} c_{n} e^{- i λ_{n} t / ℏ} u_{n} = \sum_{n \in S} c_{n} e^{- i λ_{n} t / ℏ} u_{n}$ , and $∥ ψ (t) ∥ = ∥ ψ_{0} ∥$ by the norm statement of [F1]. If all $λ_{n}$ with $n \in S$ equal a common $λ$ , then $ψ (t) = e^{- i λ t / ℏ} ψ_{0}$ , so $[ψ (t)] = [ψ_{0}]$ for every $t$ . This is (i).

2.1givenstep 1.1F2F3F4algebra

[Trigonometric-polynomial law] Assume first $∥ ψ_{0} ∥ = 1$ . By [F2] and step 1.1, using conjugate-linearity in the second argument, $μ_{[ψ (t)]}^{A} (B) = ⟨ E_{A} (B) \sum_{n \in S} c_{n} e^{- i λ_{n} t / ℏ} u_{n}, \sum_{m \in S} c_{m} e^{- i λ_{m} t / ℏ} u_{m} ⟩ = \sum_{m, n \in S} c_{n} \overline{c_{m}} e^{- i (λ_{n} - λ_{m}) t / ℏ} ⟨ E_{A} (B) u_{n}, u_{m} ⟩$ . For general nonzero $ψ_{0}$ , the unit representative $ψ (t) / ∥ ψ_{0} ∥$ has coefficients $c_{n} / ∥ ψ_{0} ∥$ by step 1.1; substituting these into the unit-vector calculation divides the displayed sum by $∥ ψ_{0} ∥^{2}$ , as asserted.

2.2givenstep 1.1F2F3F4algebra

[Return-time criterion] Let $T > 0$ . If (a) holds, then for every $t$ there is $ζ (t) \neq 0$ with $ψ (t + T) = ζ (t) ψ (t)$ ; pairing with $u_{m}$ for $m \in S$ and using orthonormality [F4] gives $c_{m} e^{- i λ_{m} (t + T) / ℏ} = ζ (t) c_{m} e^{- i λ_{m} t / ℏ}$ , hence $e^{- i λ_{m} T / ℏ} = ζ (t)$ for every $m \in S$ , which is (b). If (b) holds with common value $ζ$ , then $ψ (t + T) = ζ ψ (t)$ by step 1.1, which is (a). Finally (b) holds exactly when $e^{- i (λ_{m} - λ_{n}) T / ℏ} = 1$ for all $m, n \in S$ , and by the exponential kernel $\ker (\exp) = 2 π i Z$ of [F3] this is exactly the integrality condition (c). This proves (iii).

3.1givenstep 2.2F3algebra

[Common frequency and return time] Condition (iv) is the statement that the numbers $(λ_{n} - λ_{n_{0}}) / ℏ$ are integer multiples of one positive number $ω_{0}$ ; by step 2.2 with $T = 2 π / ω_{0}$ and [F3], $(λ_{m} - λ_{n}) T / ℏ = 2 π (k_{m} - k_{n}) \in 2 π Z$ for all $m, n \in S$ , so $T$ is a return time. Conversely, if $T > 0$ is a return time, then by step 2.2(c) applied with $m = n_{0}$ we have $(λ_{n} - λ_{n_{0}}) T / ℏ \in 2 π Z$ for all $n \in S$ , that is $(λ_{n} - λ_{n_{0}}) / ℏ = ω_{0} k_{n}$ with $ω_{0} : = 2 π / T > 0$ and integers $k_{n}$ ; this proves the equivalence of (iv).

3.2givenstep 2.2algebra

[Two energies] Let $N = 2$ , $λ_{1} \neq λ_{2}$ , $c_{1} c_{2} \neq 0$ and $T > 0$ . By step 2.2(c) with ${m, n} = {1, 2}$ the return times are exactly the solutions of $∣ λ_{1} - λ_{2} ∣ T / ℏ \in 2 π Z$ , that is $T = 2 π ℏ k / ∣ λ_{1} - λ_{2} ∣$ with $k \in Z$ , $k \geq 1$ ; the least such $T$ is $2 π ℏ / ∣ λ_{1} - λ_{2} ∣ = 2 π / ∣ ω ∣$ for $ω = (λ_{2} - λ_{1}) / ℏ$ . This is (v).

3.3givenstep 1.1step 2.1F1F2algebra

[Stationary case] If all $λ_{n}$ , $n \in S$ , equal a common $λ$ , then by step 1.1 $ψ (t) = e^{- i λ t / ℏ} ψ_{0}$ and $[ψ (t)] = [ψ_{0}]$ ; by [F2] the outcome law of every bounded assigned sharp observable depends only on the ray, so $μ_{[ψ (t)]}^{A} (B) = μ_{[ψ (0)]}^{A} (B)$ for every $t$ . This is (vi), and it agrees with the stationary-state characterization of Energy conservation, conserved sharp observables and the stationary eigenrays clause (c).

4.1step 1.1step 2.1step 2.2step 3.1step 3.2step 3.3∎

Steps 1.1, 2.1, 2.2, 3.1, 3.2 and 3.3 establish (i)-(vi) for the explicit model data.

Remarks

The relative phases are physical. Only the relative phases $e^{- i (λ_{n} - λ_{m}) t / ℏ}$ appear in the laws; a common phase of $ψ (t)$ would not change any outcome law, in agreement with the ray reading of pure preparations.
Existence of a common frequency is a number-theoretic condition on the model data. For two distinct energies it always holds, and the least return time is the inverse Bohr frequency of the precession example. For three or more energies it may fail, in which case the ray never returns to itself, although the laws remain finite sums of oscillations with the Bohr frequencies.
No detector is modeled. Exactly as on the precession example, the probabilities displayed are model outcome laws; no measurement interaction, finite resolution, efficiency or record is introduced.

A finite superposition of energy eigenrays: exact phases, laws and return times

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

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